## Problem of the Day #139: Number of tricky setsAugust 5, 2011

Posted by Mitchell in : potd , trackback

A subset $S$ of $\mathbb{Z}^3$ is called tricky if the following two conditions hold:

• For any $x, y$ in $S$ (not necessarily distinct), $x+y$ is in $S$.
• $(42, 0, 0), (0, 42, 0), (0, 0, 42), (-42, 0, 0), (0, -42, 0), (0, 0, -42)$ are all elements of $S$.

Find the number of tricky subsets of $\mathbb{Z}^3$.

Note: $\mathbb{Z}^3$ is the set of all ordered triples of integers. Additionally, if $x = (x_1, x_2, x_3)$ and $y = (y_1, y_2, y_3)$, then $x + y$ is defined to be $(x_1 + y_1, x_2 + y_2, x_3 + y_3)$.

1. phenomist - August 5, 2011

does subset here mean “proper subset”?

2. Mitchell - August 5, 2011

no

3. Mitchell - August 6, 2011

The originally posted answer to this problem was incorrect. It has been changed. We are sorry for the inconvenience.

4. Mitchell - August 6, 2011

(Note: you do not need to resubmit, because your previous submissions were saved)